Affective neuroscience, incorporating neurophysiology and neuropsychology, is providing increasing evidence that certain behaviours of animals may be interpreted in terms of what they are intending to achieve, i.e. their goals. It is also providing evidence that allows inferences to be made about the affective contents of some goal-directed behaviours. These neuroscience-supported inferences are aligned with recommendations based on prior behaviour-based investigations of animals' preferences, aversions and priorities, and these observations together support the cautious use of particular behaviours to infer what the accompanying affects may be. In this review, therefore, some attention is given to negative affects and their relationships to poor animal welfare, but the primary focus is the positive affects animals may experience when they successfully engage in rewarding goal-directed behaviours, encapsulated in the concept of positive affective engagement. The review draws together reports of environment-focused and animal-to-animal interactive behaviours observed in a range of species and under diverse circumstances in order to illustrate the likely widespread occurrence of the positive affects that may accompany them. Particular consideration is given to affects that are potentially associated with some aspects of exploration and food acquisition in stimulus rich or impoverished environments, and to those that may be associated with aspects of the affiliative interactions of bonding or bond affirmation, maternal care, play and sexual activity. It is concluded that animals given the opportunity to engage in such activities may experience some positive affects. However, the intensity of an animal's experience of particular positive affects is likely to range from zero to very high because the associated behaviours occur intermittently, variation may occur during different phases of a goal-directed behaviour, and other positive or negative affects experienced at the same time may have greater impact. As good welfare is achieved both by minimising negative affects and promoting positive ones and as conscious sentient animals may be expected to have an interest in experiencing as little pain and as much pleasure as possible, it is argued that there is an ethical obligation to take practical steps to help them to achieve these outcomes. Such steps would include providing them with opportunities to express more behaviours that are associated with rewarding or satisfying experiences understood in terms of positive affective engagement.